
These qubit outcomes can then be combined into stabilizer measurement outcomes and used in the subsequent decoding procedure of the outer code [1]}.
Additionally, we use our model for noise and the inner decoder strategy to calculate (marginal or correlated) probabilities of qubit error in our readout, which in turn can then be used to inform our outer decoder strategy that we outline in the following subsection.
The standard map from homodyne measurement outcomes to qubit measurement outcomes is a binning function derived from the translational symmetry of the original GKP state, i.e., the perfect periodicity in the \(q\)  and \(p\)  directions.
The \(\mathinner {|{+}\rangle }_{\rm gkp}\)  and \(\mathinner {|{-}\rangle }_{\rm gkp}\)  states are each \(2\sqrt{\pi }\) -periodic in momentum but shifted relative to each other by \(\sqrt{\pi }\) .
Therefore we can place the homodyne outcomes into bins of width \(\sqrt{\pi }\)  that are centred at integer multiples of \(\sqrt{\pi }\) , associating with \(\mathinner {|{+}\rangle }_{\rm gkp}\)  (\(\mathinner {|{-}\rangle }_{\rm gkp}\) ) the outcomes that fell in bins centered about even (odd) integer multiples of \(\sqrt{\pi }\) .
We refer to this procedure as “standard binning".
While this binning procedure uses the original symmetry of the GKP states, it does not account for the correlations in the covariance matrix introduced by the \(CZ\)  gates and the presence of \(p\) -squeezed states, as described in the error model.
